Multiplying mechanism



Jan. 6, 1953 E. nAwsoN MULTIPLYING MECHANISM 5 Sheets-Sheet l Filed NOV. 29, 1950 Jan. 6, 1953 E. nAwsoN 2,624,506

MULTIPLYING MECHANISM Filed Nov. 29, 1950 5 Sheets-Sheet 2 COUNTER INVENTOR 0 Wfl/P0 D/a w30/v Jan. 6, 1953 E. DAwsoN 2,524,505

MULTIPLYING MECHANISM Filed Nov. 29, 1950 5 Sheets-Sheet 3 Jan. 6, 1953 E. DAwsoN 2,624,506

MULTIPLYING MECHANISM Filed Nov. 29, 1950 5 Sheets-Sheet 4 INVENTOR Eon/Afm Dn ws 0^/ MULTIPLYING MECHAN ISM Filed NOV. 29, 1950 5 sheets-sheet 5 ANALOG m 7n rm/ERsER /wl/r *S INVENTOR EDWARD DAM/.smv

AMW 1 -A QRNEY.

'Pianeta Jan. a isss UNITEDA STATES PATENT OFFICE 2.624.506 mmmo Mscnsmsu Edward Dawson,

My invention relates generally to computing apparatus. and has reference more particularly to multiplying mech f In order to obtain more accurate results from an analog multiplying device, it is necessary to increase its physical dimensions or to improve the precision of its construction, but both of these methods of increasing the accuracy of an analog multiplier are limited in their practical applications. The attainment of extreme mechanical precision is opposed by high manufacturing costs, the difficulty of providing interchangeability of parts, deflections of the component parts, and thermal expansion thereof. On the other hand, when it is desired to secure high precision in the manufacture of an analog computing device by increasing its size, the space requirement and weight thereof often tend to be greater than the specific application within a system will permit. In general, therefore, it is not possible to construct a computing mecha- NswY NJ wim to Sperry Corporation, s nk n corporation of Delaware Appussaon November sa, 195o, semi Ns. man

` Claims. (Cl. S35-61) in which the measure 0f the nism based purely on conventional analog principles which will provide any desired degree of accuracy and therefore it is necessary in such a system to effect a compromise between the above conflicting considerations, i. e., accuracy, size,

and manufacturing cost.

Because of these` limitations of analog multipliers, it is necessary to use digital computing methods to obtain results of the desired degree of accuracy. However, the use of digital methods may necessitate the provision of an elaborate computing system containing an undesirable, large number of components so that a combined requirement for high accuracy in computing sizes may still not be satisfied. Moreover. the opera.- tion of digital computers is essentially discontinuous, consisting of a succession of individual steps, so that supplementary converting devices are necessary when it is desired to represent input and output quantities by continuously varying physical magnitudes such as shaft rotations or voltages. y

The multiplying mechanism of my invention has been devised with all of the above disadvantages of prior mechanisms considered and the resulting mechanism not only possesses the basic features and advantages of an analog multiplier, but further includes features which greatly increase its accuracy to a degree heretofore unattainable.

It is a primary object of my invention, therefore, to provide a multiplying device which has extremely high accuracy and yet may be comparatively small in physical dimensions` Another obiect of my invention is to provide a multiplying mechanism in which the inputs to a conventional analog multiplying device are recycled with continued or decreasing of the factors represented thereby beyond the capacity of the analog multiplier, the recycled product interpolated between predetermined discrete multiples of the factors. thereby providing a high degree of accuracy with a minimum size of equipment.

Still another object of my invention is to provide a multiplying mechanism-for deriving the product of a pair of factors in whichgmeasures of the first factor and a portion of the second factor. less than a predetermined fraction of the second factor, are multiplied together, Aand first factor is multiplied by the whole number of fractionate parts constituted in the second factor, and in which the products of both multiplications are added together.

A further object of my invention is to provide a multiplying mechanism for deriving a product of two input factors which comprises two multiplying means, the first of which provides a plurality of outputs each proportional to a discrete multiple of the first of the input factors, and the second of which provides an output proportional to a. continuous product of the nrst'of said factors times a second of the factors. this continuous product having a value of suillcient magnitude to include the intervals between thediscrete multiples provided by the first multiplying means, and a means for combining the results of both of said multiplications to provide an output proportional to the continuous product of the two factors.

Other objects and advantagesof my invention vwill become more apparent from the following Fig. 4 graphically illustrates the intermittent output of the reversing mechanism of Fig. 2 plotted as a function of the input factor n Fig. 5 graphically illustrates the displacement and then havingv fractionate part of the.

'of me key shifting rack plotted as a function of Fig. 'l represents schematically a multiplying device constructed in accordance with the teachings of my invention in which both of the input factors are recycled.

Referring now to Fig, 1. the factors m and n which are t be multiplied together are represented as rotations of an m-input shaft I0 and an n-input shaft I I. These shafts may be shafts of a computing system in` which the factors represented by the rotational position thereof are to be multiplied. In the embodiment illustrated in Fig. l it is assumed that the n-input shaft II requires a great number of rotations in representing the factor n, the number of rotations thereof being far in excess of the rotational limits or range of operation of one input of an analog multiplying device I2. The m-input shaft branches in two directions, one shaft I3 going t0 a plurality of multiplying means I4, I5, IB, I1 and I8, and the other shaft I9 going to the m-input of the analog multiplier device I2.

The input shaft I9 of the analog multiplier device I2 may be connected to the m-input shaft I0 through suitable reduction gearing (not shown) so that the rotational limits thereof do not exceed the limits or range of operation of the multiplying device.

The plurality of multiplying means I4'-I8' are rotated in a direction opposite to the rotavtion of the plurality of multiplying means I4-I 8,

through gear coupling 20. The outputs of the multiplying means I4'-I8' are employed when the value represented by the input factor n is negative.

The plurality of multiplying means I4-I8, I4'-I8, which, in the embodiment illustrated, comprises a plurality of gear connections of successively increasing speed ratios, provides a plurality of discrete multiples of the factor represented by the m-input shaft I0. These discrete multiples, in the embodiment illustrated, are m, 3m, 5m, 7m and 9m and -m, 3m, -5m, -7m, and 9m represented respectively by a plurality of multiplier means output shafts 2I, 22, 23, 24 and 25, and 2i', 22', 23', 24' and 25', respectively. The plurality of discrete multiple outputs are used in a manner to be hereinafter more fully described. The ninput shaft II is coupled with a converting means or modifying means 21 which, in the illustrated embodiment of my invention, comprises a reversing mechanism which provides two outputs, one being a modified version of the input factor 11.. i. e., a reversing function thereof which successively increases and decreases linearly between predetermined rotational limits. 'I'his one output ofthe reversing mechanism 21 may be' designated as a sawtooth.output" or reversing output" having a value represented by the factor 2:, graphically illustrated in Fig. 3. From the graph of Fig. 3 it will be noted that the sawtooth output a: varies linearly between values of +1 and ,-1, with reversals in slope at every even value of the input factor n. The mechanical details of a reversing mechanism which may be employed in carrying out the principles of my invention is illustrated in Fig. 2 and will be discussed in more detail below. It will be understood that the crests of the sawtooth output function :z: are not perfectly sharp, but are slightly rounded in order to 4 permit a reversal to occur with smoothness and without mechanical shock or impact. 'The sawtooth output x appears as a rotation ofoutput shaft 28 rst in one direction and then in the opposite direction within predetermined rotational limits in cach direction. This output 28 is transmitted to the analog multiplier unit I2'.

The other output of reverser mechanism 21 may be termed the intermittent output which. in the illustrated embodiment of the reverser of Fig. 2, is the rotation of the Geneva mechanism,

which output appears'on shaft 29 and rotates through a definite angle whenever a reversal occurs, in the present illustrative embodiment, through an angle of 180 at each even integral value of n. (See the graphs of Figs. 3 and 4.) Intermittent output shaft 29 rotates a pinion 30 in mesh with a rack 3l at the predetermined limits of each multiplying cycle of the analog multiplier :c-scale. The purpose of the rack and pinion 3i, 38 will be hereinafter more fully described,

An analog multiplying device which may be employed in the illustrated embodiment of my invention is shown in detail in Fig. 6 and a complete description thereof will be hereinafter set forth. For the present, however, it will be pointed out that the output shaft 32 of the multiplier I2 is rotated in accordance with the product This' isthe continuous analog product which is to be interpolated between the discrete multiples of the input factor m, that are represented by the rotational positions of the plurality of multiplier meansoutput shafts 2I-25 and 2I'-25'.

Consideration will now be given to the manner in which the continuous analog product :om is interpolated between the discrete multiples of m. The output shafts 2 I-25, 2| '--25, which respectively represent the values m, 3m, 5m, 'lm and 9m.,

and -m, -3m, -5m, -7m, and 9m are algebraically combined by means of differentials 34, 85, 36, 31 and 38, and 3', 35', 36', 31 and 38', with the output :cm of the analog multiplier I2 which rotates the spiders of the diferentials in opposite directions for positive and negative values of :vm

respectively, through bevel gearing 39. In other" words, the output 32 of the multiplying device I2 will be algebraically combined with alternate outputs of the plurality of multiplying means I 4--I8, I4-I8' in a sense corresponding to the sense of the product :rm and the output 32 of the multiplying device I2 will be algebraically combined with the remainder of the outputs of the plurality of multiplying means I4-I8, I4'-I8' in a sense opposite to the sense of the product xm. 'I'he algebraic sums or outputs of the differentials 35-38, 35'-38' are represented by rotation of differential output shafts 4D, 4I, 42,. 43 and 44, and 40', 4I 42', 43', and 44', respectively and rotate gears 45, 46, 41, 48 and 49, and 45', 45', 41', 48', and 49', respectively, which in turn engage output gears 50, 5I, 52, 53 and 54 and 50', 5I', 52', 53' and 54', respectively, which are rotatably mounted on an output member or sliding output shaft 55. Output shaft 55 has a tapered key 56 which is adapted to selectively engage suitable slots or keyways in each of the output gears Sil-54, 5'-54', in acrazcndance with the translational position of rack As the value of the input factor 11. is increased or decreased from zero, the intermittent output 29 of the reverser 21 rotates through 180 at each even integral value of n causing the rack 3i to be displaced, thereby transla-ting vkey 56 from engagementwith one of the output gears E50-54, 50'-54' to another. The rotational position of I the sliding output shaft Il' is therefore successively equal Ato the rotational positions of each of the output gears, which are. rrr-xm, :m4-xm,

7m-sm. and -Qm-i-:cm for negative values intermittent output 2l of the reverser Il. From an inspection of these graphs it will be evident that a reversal of function :c occurs at each even integral value of n (Fig. 3) at which time intermittent output sha-ft 20 rotates through 180 (Fig. 4). Such rotation produces a displacement of rack Il. to the right of, for example, 0.4 inch (Fig. 5). Thus, when the input factor n increases from a value, for example, of n= to a value of n=2, the rack Ii is in such position that the key Il is in the keyway of gear B0, therefore producing a rotation of the sliding output shaft l proportional to the product m-z1n. The notation Key in 50," Key in 51, etc. appearing on the graph of Fig. 4, is intended to indicate, with reference to the notations in Fis'. l, the product which is being delivered by the sliding output shaft 5l of the multiplying mechanism. During this interval, i. e., the interval between the values n=0 and n=2, the output shaft 2l will remain stationary and the rack Ii will remain at the above position. However, when the input factor n increases from a value of 2 to 4 (Fig. 3) the value of a: will increase from -l to +1 passing through zero when n equals 3. At the reversal i. e., when n=2, the intermittent output shaft 29 will rotate through 180 (Fig. 4) thereby shifting rack Il furtherto the right causing the key 56 to be transferred from engagement with output gear 5I to engagement with output gear 5|. At the completion of the reversal, intermittent output shaft 28 will again be stationary causing rack Il to remain stationary so that as the value of n increases from 2 to 4 the rotation of output gear 5| will be equal to the product 3m+a:m. In the illustrated embodiment of my invention, the graphs indicate that this laction *continues through positive values of input factor n up to the value n=10.

Consideration will now be given to the actuation of rack 3| when n decreases in value from n=0 to 11L=10. When the input factor n decreases from the value of 0 to -2 the value of .1: decreases from +1 to -1 but the slope of the line representing .r is opposite to the slope of the line when n increases from zero to +2. From an inspection of the graph of Fig. 5, it will be observed that the displacement of rack 3l will be exactly equal to what it was when the value of n increased from n=0-to n=+2, but the rotation of the intermittent output shaft 29 will be in the opposite direction and therefore the translation of the rack 3i will be to the left, thus translating key I6 from engagement with the output gear 50 to output gear $0'. Therefore, when n' decreases in value from n=0 to n=2 the product output of the sliding output shaft 5B will be equal to m+xm (Fig. 4). As n decreases further in value i. e., from n=2 to 11:-4, :r will increase in value from -l to +1 passing through 0 at the value considering a few numerical examples.

accesos =3. However, at n=2 the intermittent output shaft II will again rotate through thereby shifting rack Il further to the left so that key Il will be transferred from engagement with output gear Il' to output gear ll', where the rack will remain until n equals 4. During this lnterval the output shaft Il will be rotated in accordance with the product 3m-xm. In the illustrated embodiment of my invention, this action will continue for negative values of n from n=0 t0 1l=-10.

From the above discussion it will be seen that the integral multiples provided by the multiplying means I4-II will provide a plurality of discrete multiple outputs of successively increasingv or decreasing rotational values, and that the product xm is algebraically combined with each of these multiples. However, the rack and pinion Il, l0, which is controlled by predetermined values of n, determines which product appearing on output shafts Ill-44, 44' will be connected to rotate the output shaft It. In this manner the factor n will be included in the rotation of the output shaft l0 and its rotation therefore will equal to the product mn.

This continuous product mn may be coupled with a further output shaft through an elongated pinion 59 engaging spur gear Il mounted on the end of sliding output shaft Il which, in turn, is in engagement with the rack 3 I.

AsA above indicated with respect to Fig. 5, in a practical application of the multiplying mechanism of the present invention the intermittent translational travel of the'rack Il may be, for example, 0.4 inch per step of 180 input rotation. Under such conditions, the output gears "-34, BIV-54 would be located at 0.4 inch center to center distance measured along the sliding output shaft 5l. The width of each of these gears, measured at the hub, would be 0.4 inch, less a small amount for running clearance, so that the idler gears form a compact assembly instead of being widely separated as shown in the drawings for the purposes of illustration.

It will be noted that the input n, as it continuously increases or decreases, is converted or modified by the reversing mechanism 21, to a successively increasing and decreasing function thereof, the rotation of the output shaft representing this function not exceeding the range of operation of the :r-scale of the multiplying device I2, and that this function is multiplied with the other input factor m to provide an incremental product xm. However, the m-input is simultaneously multiplied in large steps to provide outputs proportional to discrete multiples thereof. 'Thereafter the incremental product rm, provided by the analog multiplier i2, is interpolated by addition and subtraction between the discrete multiple outputs of m to provide a continuous product of the two input factors mand n. Of the total tunis of the output shaft 55, which represents the product rm, only a very small fraction thereof is contributed by the analog multiplier I2, the rest being obtained from the continuously rotating gears of the plurality of multiplying means. Thus, the accuracy of the ana` log multiplier may be increased many times by the recycling mechanism above described.

The operation of the multiplying mechanism illustrated in Fig. l will be better understood by It will be assumed that initially ms- O and 11:0. From an inspection of the graph of Fig. 3, it will be seen that for the condition 11:0, the value of :c is

+1, hence Athe initial value of the output, mp1., of the analog multiplier 12 is zero. Moreover, under such conditions the output, at shaft 2|, of the first of the plurality of multiplying means I4-I3 equals zero and therefore the 'rotational position of output shaft 55 represents a zero output. Now assume that m increases to, say. a value of +1; the value of n remaining zero. Under these conditions the output :cm of multiplier I2 is equal to +1 andthe output of the multiplying means I4 at shaft 2I also equals +1. However, since the output of multiplier I2 is reversed in sign by one of the pairs of bevel gears 39 before it is algebraically added. at differential 34, to the output of multiplying means I4, the rotation of output shaft 40 still represents a value of zero, and the value represented by output shaft Il is also zero. Now, if the value of m remains +1 and the value of n also increases to +1, the value of a: is equal to 0, as illustrated by the graph of Fig. 3, and the output ma: of the multiplier I2 will decrease to zero. Therefore, since the output of the first multiplying means I4 remains equal to +1, the rotational position of the out- -put shaft 4l) of the differential 34 will equal +1.

Assume now that the value of m remains equal to +1 but the value of n is increased to a value of 1.5. In such case :c becomes equal to 0.5 (Fig. 3) and the output m of the multiplier device I2 will equal 0.5, since m equals +1. Since the output 1n of the multiplying means I4 at shaft 2| is +1, and the product :rm is 0.5, the output 40 of the differential 34 is equal to m-m or 1 0.5) or +1.5. This is the product which appears as a rotation of output shaft 55. Now let n increase to a value of +2.5 and the value of m equals +0.76. From the graph of Fig. 3, when n=+2.5, :cr- 0.5. When, in increasing to 2.5, the value of n is equal to 2, a reversal of the reverser 21 occurs as hereinabove described, thereby recycling the output of the analog multiplier I2 and at the same time shifting the key 56 from engagement with idler gear 50 to engagement with idler gear I. 'I'he output xm of the analog multiplier I2 is then equal to (-0.5) (+0.76) 0.38. However the output of multiplying means I5 which drives idler gear 5I through shaft 22, differential 35, shaft 4I, and gear 46, is equal to the product of m and the discrete factor +3, and its output shaft 22 is rotated in proportion to +3m. Since the output :vm of the multiplier device I2 is applied to the differential 35 without a reversal of signoutput of diierential 35, at shaft 4I, will represent the value 3m+m or This value is represented by the rotational position of output shaft 55 since the key 56 is now in engagement with the idler gear 5I.

Let us consider another example in which the values of the factors m and n are fractionate numbers. Assume that the factor n increases to a value of +8.3 and that the value of m equals +0.76. From an inspection of the graph of Fig.

' 3, representing the output of the reverser 21, it

is apparent that when the value 'of n equals +83,

:n is equal to +0.7. Since the input :c to the mul-1 8 the output of multiplying means I6, the output of differential 38 will represent 'a value of This is the product which will appear as a rotation of differential output shaft 44. idler gears 49 and 64, and hence output shaft 55.

The above described sequence of operations will occur for both positive and negative values of inputs m and n, but in the case of negative values, the rotations are reversed and the key 56, instead of being moved to the right will be moved to the left, thus coupling the sliding output shaft 55 with gears 45'-49', differentials 3:'-38', and the plurality of multiplying means I -I8',

In the preceding discussion of the multiplying mechanism of myv invention, I have referred generally to the reversing mechanism 21 in Fig. 1. 'I'his device is similar in principle to that described in my copending application Serial No. 103,465, filed July '1, 1949, for an Automatic Reversing Mechanism, and the mechanical arrangement illustrated in Fig. 2 of the present application is substantially identical to the arrangement of the reversing mechanism illustrated in Fig. 5 of this copending application. However, in order to further a better understanding of the present invention I will herein describe this mechanism in detail.

When the n-input shaft II of the multiplier is rotated continuously in either direction, the output shaft Z8 undergoes the periodic reversals represented graphically in the diagram of Fig. 3, The reversing cycles are initiated by a counter mechanism 65 driven, through suitable gearing connections, by the output shaft 28 of the reverser. The main difference between the reverser disclosed in the above-mentioned application is that in the latter the counter is driven by rotation of the input shaft rather than rotation of the output shaft. The rotation of the counter output shaft 66, produced by the sawtooth rotation of its input, rotates through 190 from its central or neutral position, and

by means of a 2 to 1 gear ratio connection 61, turns a crank 68 through an angle of -180. This crankl 68 is arranged to slide the star Wheel 69, by means of sliding yoke 64, of a Geneva drive device 10 from its normal posi,- tion of engagement with fixed detent member 63 which is axially displaced from rotating member 12, into axial alignment with the Geneva driving member 12. Thus, the rotation of the counter output shaft 66 through either plus or minus will move star wheel 69 into an axial position Where it will be actuated by a Geneva drive member 12. This axial shifting takes place when the pin 1I of member 12 is clear of star wheel 69. Drive member 12 is geared to rotate at one half the speed of the reverser input shaft Il. When the counter 65 has moved the star Wheel 69 into engagement with drive member 12 the consequent rotation of star Wheel 69 through an angle of 90 is multiplied to 180 by gears 13 of 2 to 1 ratio on the Geneva output 14. This rotation is transmitted to cranks 15 and 16 through suitable bevel gearing 11, and to the input shaft 13 of a Scotch yoke device 19 through suitable bevel gearing 80, As will become apparent as a description of the reversing mechanism proceeds, a reversal of the reverser output shaft 28 is effected when this 180 rotation of Geneva output shaft 14 occurs. The intermittent rotational motion of shaft 14 is made 9 available, by means of gearing 14', as an "intermittent outpu at shaft 29.

- The reverser output shaft 28 is normally driven by gears 8| or 82, which are driven by input gear 83 directly connected to input shaft Il, and oppositely rotating gear 84 in mesh therewith, through suitable jaw couplings 85 and 88 respectively. When a reversal takes place, drive of the output shaft 28 is shifted from gear 8| to gear 82, or vice versa, by engaging and disengaging the appropriate couplings 8B, 88 and 86, 86'. If the drive operates through coupling 25, 85' at the beginning of the reversal, then coupling half 90 is made to match coupling half the output 88 of Scotch yoke device 19 has moved the rack 81 to one of the extreme positions of its stroke, the output shaft 28, driven by coupling 90, 90', is brought to rest. Rack 81 then starts back towards its neutral position, and when the speeds and angles of coupling halfs Bland 86' are equal, the drive to the output shaft 28 shifts from 90, 90' to 86. 86' by rotation of crank 16, rocker arm 16 and cylindrical translating cam member 89. The rack 01 then comes to rest at its neutral position and the output shaft 28 runs in the opposite direction. The counter output shaft 66 is thereupon turned in the opposite direction through 90, thereby disengaging drive wheel 12 and star wheel 69, thus completing a reversing cycle.

In Fig. 6 I have schematically illustrated an analog multiplying device which is well suited to the multiplying mechanism of my invention. It is extremely simple both in operation and in construction. This multiplying device operates on the principle of prosthaepheresis using square lfunctions in accordance with the equation:

In this multiplying device the input shafts and |0|, which correspond respectively with reverser output shaft 28 and m-input shaft |0 of Fig. 1, are rotated in proportion to the factors s: and m respectively. Factor is supplied through suitable bevel gearing |02, |03 in a positive sense to a pair of input differentials |04 and |05, respectively. The other input m is supplied positively to the differential |04 through bevel gearing |06 and |01 and negatively to the input of the other dierential through bevel gearing |06, and |08. The output of differential |04 therefore will appear as a rotation of shaft |09 proportional to the sum (x4-m) and similarly the output shaft I|0 will I be rotated in proportion to the quantity (x-m).

configuration such that as they are respectively rotated in proportion to the quantity (x4-m) and (-ml the follower sectors IIS and H6 are rotated in proportion to the quantities (:+m)= and (1r-1m respectively. The difl0 ference in rotation of sectors IIB and H8 is obtained by an outputdifferential |2| having an output shaft |22 corresponding to shaft I2 of Fig. 1, which is rotated in accordance with the product 4am. In a practical application, the construction of this multiplying device can be made quite simple. The shafts for the various pinions, cams, and differentials can be supported by fiat, parallel, side plates'which are connected by and suitably attached to spacer blocks. The cams a'nd ||2 may be duplicates and may be assembled back to back between the side plates and the sector gears |I5 and H6, which may also be duplicates. may be suitably mounted for free rotation on shafts of the two input differentials and the free ends thereof may be supported by slots in the spacer blocks. The differentials |04 and |05 may be journaled between the side plates and instead of having output shafts connected to their spiders, as illustrated, suitable gear teeth may be provided directly on the surface of the spider which gear teeth may drivably engage the toothed periphery of calms and H2. Simif larly, the output differential |2| instead of having an output shaft |22, as illustrated, may also have gear teeth fabricated directly in the surface of the spider and an output pinion in mesh therewith.

In the above description of the multiplying device of my invention I have illustrated a mechanism in which the :r-input of the basic multiplier or analog multiplier I2 is increased cyclically for continued increases in the value n but in which the m-input is not recycled. In a practical application of the multiplier mechanism of my invention, it would be desirable to increase both scales of the analog multiplier, i. e., increase the accuracy of both factors n and m. This may be accomplished by substituting for the analog multiplier I2 of Fig. 1, the entire recycling multiplier of Fig. 1 to obtain thereby a continuous product rm. In doing this the input m is converted to a sawtooth function by. a reversing mechanism which may be substantially identical to the n-reversing mechanism 21 of Fig. l and the scale factor of variable m may be enlarged as many times as required. Such a system is schematically illustrated in Fig'. 7.

In Fig. 7 the m input I0 branches in two directions, one shaft I3 going to the plurality of multiplying means |4 through I8 and the other shaft I9, instead of going directly to the analog multiplier |2 as in Fig. 1 goes to an mreversing mechanism |29. As before, the purpose of the multiplying means |4-I8 is to provide a plurality of discrete multiples of the shaft rotation represented by the input factor m. The m-reverser |29 is substantially identical to the n-reverser 21 and produces a pair of outputs, one of which appears on output shaft |20 and is designated as the sawtooth output 1l and the other output appearing on shaft |2| which is designated as the intermittent output. The sawtooth function y is exactly the same as sawtooth function a: of Fig. 1 and varies linearly between values of +1 and .-f1 withreversals in slope at every even value of m. The sawtooth output y appearing on shaft |20 is coupled with the 1l-input of en analog multiplying device |32 which may be identical to the multiplying device illustrated in Fig. 4. Again, the intermittent output III of vthe mreverser rotates through an angle of whenil Y ever a reversal occurs; that is, at each \even integral value of m. This shaft translates the key shifting rack |33 at the limits of eachbycle,

or at the limits of the range of operation of the Y analog multiplier :Ll-input.

Returning now to a consideration of the nt v and the other of which enters, by way of shaft 28', the analog multiplier device |32. The vanalog multiplier device |32 provides a rotation of its output shaft |40 which represents the product :cy of its two input factors. 'I'his is the continuous analog product which is to be interpolated between the discrete multiples of input4- factors :c and m. The intermittent output 29 of the n-reverser 21 translates the lower key shifting rack` |4| through 180 whenever sawtooth function :c reverses.'

Consideration will now be given to the manner in which the continuous analog product my is interpolated between the discrete multiples of function which are provided by the upper set of multiplying means |35|39. As illustrated in Fig. 7, the upper set of multiplying means receive sawtooth output :c and deliver five outputs zc, 3x, 61:, 1a: and 91: and are represented by rotations of a plurality of multiplying means voutput shafts |43-, |44, |45, |46 and |41, respectively. 'Ihese outputs are combined, by means of ve differentials |50, |5I, |52, |53'and |54 with the output :ry of the analog multiplier |32 which rotates the spiders of the differentials in proportion to -I-:cy or 3:11. The sums of the outputs of the plurality of multiplying means and the output ofthe multiplier |32 are obtained from these differentials and rotate output gears or output members |55|59 on a sliding shaft |60 having a key |6| which is adapted to selectively engage key-ways |55'-|59' in the respective output gears. As the value of the input factor m is increased or decreased, the intermittent output |3| of m-reverser |29 rotates through 180 at each integral even value of m, causing the key-shifting rack |33 to slide key 6| from one of the output gears |55|59 to another. The rotational position of the sliding shaft |60 is therefore successively equal to the rotational positions of each of the output gears, which .are :c-y, 3+yx, 51e-yz, '1m-H13: and

. l2 vided with a key |86 and is positioned axially by key-shifting rack |4| .driven from the Geneva output 29 of the n-reverser. In this way rotation of the second sliding output shaft |85 represents the product mn. This output appears as a rotation of shaft |90 by way of a longpinion |9| and meshing gear |92 fixed to the sliding output shaft |85.

The multiplying mechanism as illustrated in Fig. 7 will accept only positive values of the 'input vfactors m and n, the additional members required for accepting negative values being omitted for the sake of clarity. However, these additional members are fully described and illusf trated in `Fig. 1 and it is to be understood that they may be 'included in the mechanism of Fig. 7, if required.

The accuracy of the cyclic multiplier may be greatly increased beyond that of the analog multiplier which it contains, because he analog -ing gears of the plurality of multiplying means.

In a practicalapplication, the accuracy of a multiplying mechanism embodying the principles of my invention is limited only by considerations of sizeI cost and the time required for deriving the product. l Y

Since many changes could be made in the above construction and many apparently. Widely different embodiments of this invention could be made without departing from the scope thereof, it is intended that all matter contained in the above ldescription or shown in the accompanying drawings shall be interpreted as illus trative and not in a limiting sense.

What is claimed is:

l. In a multiplyingv mechanism comprising a rst input member adapted to be actuated in accordance with a first factor, a second input member adapted to be actuated in accordance with a second factor, modifying means coupled with said Qca-yz. 'Ihus the rotational position of the sliding output shaft represents product zum.

As in the case of Fig. 1, a long, non-sliding, pinion |62 is rotated in proportion to the product :cm by the output gear |63 of the sliding output shaft |60, and this rotation is applied, through suitable shaft |64, as an input to drive the spider shafts |65 and |66 of a second group of five differentials |61, |68, |69, |10 and |1|. These differentials algebraically combine the quantity +:cm or -xm to the discrete multiple outputs m. 3m, 5m, 1m and 8m appearing as rotations of output shafts |12, |13, |14, |15 and |16, respectively of the plurality of m multiplying means I4, l5, I6, |1 and I8. The output shafts of the ive differentials |61|1| drive idler gears |80, |8|, |82, |83 and |84 on a second sliding first input member for producingan output proportional to a function of said first factor, the output thereof successively increasing and decreasing between predetermined limits, a multiplyingdevice having one input connected to receive the output of said modifying means and the other input connected to said second input member, a'plurality of multiplying means connected to the second input member for providing a plurality of outputs each proportional to a discretel multiple of the factor represented thereby, means for algebraically combining the output of said multiplying device with alternate outputs of said plurality of multiplying means in a sense corresponding to the sense of the output of said multiplying device, and means for algebraically combining the output of said multiplying device with the remainder of the outputs of said plurality of multiplying means in a sense oppositel multiplying 13 other input connected to said second input member, a plurality of multiplying means connected to the second input member for providing a plurality of outputs each proportional to a discrete multiple of the factor represented thereby, means for algebraically combining the output of said multiplyingdevice with alternate outputs of said plurality of multiplying means in a sense corresponding to the sense of the output of said multiplying device, means for algebraically combining the output of said multiplying device with the remainder of the outputs of said plurality of multiplying means in a sense opposite to the sense of the output of said multiplying device, an output member. and means controlled by said modifying means for selectively coupling anyone of said combining means with said output member. 3. In a multiplying mechanism comprising a .first input member adapted to be actuated in accordance with a first factor, a second input member adapted to befactuated in accordance with a second factor, modifying means coupled Awith said first input member for producing an output proportional to a function of saidviirst factor, the output thereofv cyclically increasing and decreasing linearly between predetermined limits with continuous increasing of said first factor, a multiplying device having one input connected to receive the output of said modifying means and the other input connected to said second input member. a plurality of multiplying means connected to the second input member for providing a plurality of outputs each proportional to a discrete multiple of the factor represented thereby, means for algebraically combining the output of said multiplying device with alternate outputs of said-plurality of multiplying means in a sense first factor, the

. 14 5. A multiplying mechanism comprising, a first input member adapted to be actuated in vaccordance'with a first factor, a second input member adapted to be actuated in accordance with a second factor, converting means coupled with said first input member for producing an output proportional to a output thereof cyclically increasing and decreasing linearly between predetermined limits, a multiplying device having one input connected to receive the output of said converting means and the other input connected to said second input member, a plurality of multiplying means connected to said second inputmember for providing a plurality of outputs proportional to discrete multiples of the factor represented thereby, means for algebraically combining the output of said multiplying device with alternate outputs of said plurality of multiplying means in a sense corresponding with the sense of the output of said multiplying device, means foralgebralcally combining the output of said multiplying device with the remainder of the outputs of said plurality of multiplying means in a sense opposite to the sense of the output of said multiplying device, a plurality of output members coupled with the outputs of said combining means, an output for said multiplying mechanism, and means controlled by said converting means for selectively engaging any one of said plurality of output members with the output of said multiplying mechanism.

6. In a system for multiplying a pair of factors comprising a first input member adapted to be corresponding to the sense of the output of said multiplying device, and means for algebraically combining the output of said multiplying device with the remainder of the outputs of said plurality of multiplying means in a sense opposite to thesense of the output` of said multiplying device.

4. A'multiplying mechanism comprising a first input member adapted to be actuated in accordv ance with a first factor, a second input member adapted to be actuated in accord-ance with a secY ond factor, modifying means coupled with said first input member for producing an output proportional to a function of said first factor, the output thereof cyclically increasing and decreasiineariy between predetermined limits with continuous increasing of said first factor. a multiplying device having one input connected to receive the output of said modifying means and the other input connected lto said second input member, a plurality of multiplying means connected to the second input member for providing a plurality of outputs each proportional to a discrete multiple of the factor represented thereby, means for algebraically combining the output of said multiplying device with alternate outputs of said plurality of multiplying means in a sense corresponding to the sense of the output of said multiplying device, means for algebraically combining the output of said multiplying device with the remainder ofthe outputs of said plurality of multiplying means in a sense opposite to the sense of the output of said multiplying device, an output member, and means controlled by said modifying means for selectively coupling any one of said combining means vwith said output member at each increase or decrease of the output of said modifying means.

actuated in accordance with a first of said factors, a second input member adapted to be actuated in -accordance with the second of said factors, a reversing mechanism coupled with said first input member for producing an output proportional to a reversing function of said first factor, said reversing function increasing and de creasing linearly .between predetermined limits with continued increasing of said first factor, a multiplying device having one input connected to receive said reversing function and the other input connected to said second input member, the range of operation of said multiplying device as to said one input being less than the maximum range of actuation of said first input member but not exceeding the limits of said reversing function, a lplurality of multiplying means connected to the second input member for providing a plurality of outputs each proportional to a discrete multiple of thev factor represented thereby, means for algebraically combining the output of said multiplying device with alternate outputs of said plurality of multiplying means in a sense corresponding with the sense of the output of the said multiplying device, means for algebraica-lly combining the output of said multiplying device with the remainder of the outputs of said plurality of multiplying means in a sense opposite to the sense of the output of said multiplying device, an output member, and means actuated during each reversal of said reversing mechanism for selectively engaging any one of said combining means with said output member.

7. In a system for multiplying a pair of factors comprising a first input member adapted to be actuated in accordance with a rst of said factors, a second input member adapted to be actuated in accordance with the second of said factors, a

- reversing mechanism coupled with said first inpredetermined function of said l ing linearly between predetermined limits with continued increasing of said rst factor, a multiplying device having one input connected to receive said reversing function and the other input connected to said second input member, the v range of operation of said multiplying device as tosaid one input being less than the maximum range of actuation of said first input member but not exceeding the limits of said reversing function, a. plurality of multiplying means connected to the second input member for providing a plurality of outputs each proportional to a discrete multiple of the factor represented thereby, a first group of differential means for algebraically combining the output of said multiplying device with alternate outputs of said plurality of multiplying means in a sense corresponding with the sense of the output of said multiplying device, a second group of differential means for algebraically combining the output of said multiplying device with the remainder of the outputs of said plurality of multiplying means in a sense opposite to the sense of the output of said multiplying device, an output member, and means actuable during each reversal of said reversing mechanism for selectively coupling the output of any one of said differential means with said output member.

8. A multiplying mechanism comprising a first input shaft adapted to be rotated in accordance with a first factor,.a second input shaft adapted to be rotated in accordance with a second factor. a reversing mechanism coupled with said first shaft and having an outputshaft rotatable in accordance with a continuously reversing function of said first factor, said output shaft successively rotating first in one direction and then in the opposite direction between predetermined rotational limits with continuous rotation in one direction of said first input shaft, a multiplying device coupled with said reversably rotating shaft and said second input shaft and having an output shaft rotatable in accordance with the product thereof, the range of operation of said multiplying device as to rotational output of said reversing mechanism being less than the maximum range of rotation of said first input shaft but greater than the rotational limits of the output shaft of said reversing mechanism, a plurality ofy gea-r connections coupled with the second input shaft having a -plurality of outp'ut shafts, the rotational output of each being proportional to a discrete multiple of the factor represented by said second input shaft, a first group of differentials for algebraically combining the rotational output of said multiplying device with alternate outputs of said gear connections in a sense corresponding with the sense of the output of said multiplying device, a second group of differentials for algebraically combining the rotational output of said multiplying device with the remainder of the outputs of said gear connections in a sense opposite to the sense of the output of said multiplying device, a plurality of output members driven in accordance with the output of said differentials-an output shaft adapted to be rotated in accordance with the 4product of said rst and second factors, and means operable during each reversal of said reversing mechanism for selectively engaging any one of said plurality of output members with said last-mentioned output shaft.

9. A multiplying mechanism as set forth in claim 8 in which the plurality of output mem- 16 bers are rotatably mounted on said output shaft. a key secured to said output shaft and adapted to be selectively engaged with any one of said output members, and the means controlled by said reversing mechanism comprises a rack and pinion actuated during each reversal of said reversing mechanism for shifting said keyfrom engagement with one of said plurality of output members into engagement with another of said plurality of output members. l

l0. In a multiplying mechanism comprising a first input member adapted to be actuated in accordance with a first factor, e. second input member adapted to be actuated in accordance with a second factor, first and second modifying means coupled with said first and second input members respectively for producing outputs proportional to a function of said first .and second factors respectivelyl the outputs thereof successively increasing and decreasing between predetermined limits, a multiplying device having one input connected to receive the output of said first modifying means and the other input con nected to receive the output of said second modi= fying means, a. first plurality of multipying means connected to the output of said first modifying Y modied factor, first combiningmeans for algei braically combining the output of said multiplying device with alternate outputs of said first plurality of multiplying means in a sense corresponding with the sense of the output of said multiplying device and for combining the output of said multiplying device with the remainder of the outputs of said first plurality of multiplying means in a sense opposite to the sense of the output of said multiplying device, a first output member, means controlled by said second modifying means for selectively coupling any one of said first combining means with said first output member, a second plurality of multiplying means connected to the second input memberfor providing a second plurality of outputs each proportional to a discrete multiple of the factor represented thereby, second combining means for algebraically combining the product represented by said first output member with alternate outputs of said second plurality of multiplying means in a sense corresponding with the sense of said first output member and for algebraically combining the product represented by ,said first output member with the remainder of the outputs of said first plurality of multiplying means in a sense opposite to the sense of said first output member, a second output member,l and means controlled by said first modifying means for selectively coupling any one of said second combining means with said second output member.

EDWARD. DAWSON.

lREI-Exteriorss CITED The following references are ofrecord in the file of this patent: 4

UNITED STATES PATENTS Number Name Date .1,912,857 Pinkel et a1 June 6, 1933 2,472,097 Doersam, Jr June '7, 1949 FOREIGN PATENTS Number. Country Date 340,013 Great Britain Dec. 12, 1930 v 

